Determinants and Their Applications in Mathematical Physics

122 4. Particular Determinants

Theorem 4.36.

D(T

(n,r)

11

)=−(2n+r−1)T

(n,r+2)

n,n− 1

.

Proof.

T

(n,r)

11

=T

(n− 1 ,r+2)

.

The theorem follows by adjusting the parameters in Theorem 4.35.

Both these theorems are applied in Section 6.5.3 on the Milne–Thomson

equation.

4.9.3 Determinants with Simple Derivatives of All Orders

LetZrdenote the column vector with (n+ 1) elements defined as

Zr=

[

(^0) rφ 0 φ 1 φ 2 ···φn−r

]T

n+1

, 1 ≤r≤n, (4.9.15)

where 0rdenotes an unbroken sequence ofrzero elements andφmis an

Appell polynomial.

Let

B=

∣

∣Z

1 C 0 C 1 C 2 ···Cn− 1

∣

∣

n+1

, (4.9.16)

whereCjis defined in (4.9.5). DifferentiatingBrepeatedly, it is found that,

apart from a constant factor, only the first column changes:

D

r

(B)=(−1)

r

r!

∣

∣

Zr+1C 0 C 1 C 2 ···Cn− 1

∣

∣

n+1

, 0 ≤r≤n− 1.

Hence

D

n− 1

(B)=(−1)

n− 1

(n−1)!φ 0

∣

∣

C 0 C 1 C 2 ···Cn− 1

∣

∣

n

=(−1)

n− 1

(n−1)!φ 0 |φm|n, 0 ≤m≤ 2 n− 2

= constant;

that is,Bis a polynomial of degree (n−1) and not (n
2
−1), as may

be expected by examining the product of the elements in the secondary

diagonal ofB. Once again, the loss of degree due to cancellations isn(n−1).

Exercise

Let

Sm=

∑

r+s=m

φrφs.

This function appears in Exercise 2 at the end of Appendix A.4 on Appell

polynomials. Also, let

Cj=

[

Sj− 1 SjSj+1···Sj+n− 2

]T

n

, 1 ≤j≤n,

K=

[

• S 0 S 1 S 2 ···Sn− 2

]T

n

,

E=|Sm|n, 0 ≤m≤ 2 n− 2.